Concept: Box-and-Whisker Plots

A box-and-whisker plot is a visual representation of how the data is spread out and how
much variation there is. The main advantage of the box-and-whisker plot is that it is not
cluttered by showing all the data values. It highlights only a few important features of
the data. Therefore, the box-and-whisker plot makes it easier to focus attention on the
median, extremes, and quartiles and comparisons among them. Another advantage of the
box-and-whisker plot is that it does not become more complicated with more data values.
A disadvantage of the box-and-whisker plot occurs when there are only a few data values.

Constructing a box-and-whisker plot:

Jan

Feb

Mar

Apr

May

June

July

Aug

Sept

Oct

Nov

Dec

Seattle, WA

46

51

54

58

64

70

74

74

69

60

52

46

San Antonio TX

61

66

74

80

85

92

95

95

89

82

72

64

New York, NY

38

40

50

61

72

80

85

84

76

65

54

43

First, arrange the data in the table above in increasing order.

Seattle, WA : 46, 46, 51, 52, 54, 58, 60, 64 ,69, 70, 74, 74

San Antonio, TX: 61, 64, 66, 72, 74, 80, 82, 85, 89, 92, 95, 95

New York, NY: 38, 40, 43, 50, 54, 61, 65, 72, 76, 80, 84, 85

Second, find the extremes. The lower extreme is the lowest value and the upper extreme is
the highest value.

The lowest value for Seattle is 46, and the highest value is 74.

Third, find the median.

There are 12 values, therefore the median is halfway between the 6th and the 7th values.

46, 46, 51, 52, 54, 58

| 60, 64, 69, 70, 74, 74

The median is 59. There are six numbers below 59 and six numbers above it.

Fourth, find the lower quartile (the median of the lower half of the data).

Consider only the data values to the left of the line.

46, 46, 51, 52, 54, 58

The median of these six numbers is halfway between the 3rd and the 4th values. This is the lower quartile. The lower quartile is 51.5.

Fifth, find upper quartile (the median of the upper half of the data).

Consider only the data values to the right of the line.

60, 64, 69, 70, 74, 74

Find their median. This is the upper quartile. The upper quartile is 69.5.

Sixth, plot the extremes (lower extreme 46, upper extreme 74), the quartiles (lower
quartile 51.5 and upper quartile69.5), and the median (59) on a number line.

Seventh, draw a rectangular box extending from the
lower quartile to the upper quartile. Indicate the
median with a vertical line extending through the box.

Eighth, connect the lower extreme toto the lower
quartile with a line (one "whisker") and the upper
quartile to the upper extreme with another line (the
other "whisker".)

Repeat steps one through eight to construct box-and-whisker plots using the data given for San Antonio and New York, respectively.

Note:

If there are an odd
number of data values,
the overall median is
not included among the
values used to calculate
the medians of the
lower or upper
quartiles.